Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{q^2 + 18q + 80}{q^2 + 17q + 72}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 + 18q + 80}{q^2 + 17q + 72} = \dfrac{(q + 10)(q + 8)}{(q + 9)(q + 8)} $ Notice that the term $(q + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 8)$ gives: $r = \dfrac{q + 10}{q + 9}$ Since we divided by $(q + 8)$, $q \neq -8$. $r = \dfrac{q + 10}{q + 9}; \space q \neq -8$